Over the past few years numerous estimators have been proposed to estimate the quadratic OT distance/maps. However, either these estimators can be computed in polynomial time w.r.t the number of samples yet they suffer the curse of dimension either, under suitable smoothness assumptions, they achieve dimension-free statistical rates yet they cannot be computed numerically. After proving that the cost constraint in smooth quadratic OT can be written as a finite sum of squared smooth function, we use the recent machinery of kernel SoS to close this statistical computational gap and we design an estimator of quadratic smooth OT that achieves dimension-free statistical rates and can be computed in polynomial time. Finally, after proving a new stability result on the semi-dual formulation of OT, we show that our estimator recovers minimax rates for the OT map estimator problem.